Question

1. which expression is equivalent to \\(\frac{10q^8w^7}{2w^3} \cdot \frac{4(q^6)^2}{w^{-5}}\\) for all values of q and w where the expression is defined?
2. the expression \\((x^3)(x^{-17})\\) is equivalent to \\(x^n\\). what is the value of n?

Explanation:

Question 5

Step 1: Simplify coefficients and exponents of \( q \) and \( w \) separately

First, simplify the coefficients: \(\frac{10}{2} \times 4 = 5 \times 4 = 20\).

For the \( q \)-terms: We have \( q^8 \) and \( (q^6)^2 \). Using the power of a power rule \((a^m)^n = a^{mn}\), so \((q^6)^2 = q^{12}\). Then, when multiplying \( q^8 \) and \( q^{12} \), we use the product of powers rule \( a^m \times a^n = a^{m + n} \), so \( q^8 \times q^{12} = q^{8+12}=q^{20}\).

For the \( w \)-terms: We have \( w^7 \) in the numerator of the first fraction, \( w^3 \) in the denominator of the first fraction, and \( w^{-5} \) in the denominator of the second fraction. When dividing, we use the quotient of powers rule \( \frac{a^m}{a^n}=a^{m - n} \), and when there is a negative exponent in the denominator, it moves to the numerator with a positive exponent (\( \frac{1}{a^{-n}}=a^n \)). So for the \( w \)-terms: \( w^7\div w^3\times w^{5} \) (since \( \frac{1}{w^{-5}} = w^5 \)). Using the quotient rule first: \( w^{7-3}=w^4 \), then multiplying by \( w^5 \): \( w^4\times w^5 = w^{4 + 5}=w^9 \).

Step 2: Combine all simplified terms

Now, multiply the simplified coefficient, \( q \)-term, and \( w \)-term together: \( 20\times q^{20}\times w^9 = 20q^{20}w^9 \).

Step 1: Apply the product of powers rule

The product of powers rule states that for any non - zero real number \( a \) and integers \( m \) and \( n \), \( a^m\times a^n=a^{m + n} \).

We are given the expression \( (x^3)(x^{-17}) \). Here, \( a = x \), \( m = 3 \), and \( n=- 17 \).

Using the product of powers rule, we add the exponents: \( 3+(-17)=3 - 17=-14 \).

So \( (x^3)(x^{-17})=x^{-14} \), and since this is equivalent to \( x^n \), then \( n=-14 \).

Answer:

\( 20q^{20}w^9 \)

Question 6